An elementary introduction to the non-perturbative renormalization group is presented mainly in the context of statistical mechanics. No prior knowledge of field theory is necessary. The aim is this article is not to give an extensive overview of the subject but rather to insist on conceptual aspects and to explain in detail the main technical steps. It should be taken as an introduction to more advanced readings.Comment: 56 page

This article is devoted to the study of the critical properties of classical XY and Heisenberg frustrated magnets in three dimensions. We first analyze the experimental and numerical situations. We show that the unusual behaviors encountered in these systems, typically nonuniversal scaling, are hardly compatible with the hypothesis of a second order phase transition. Moreover, the fact that the scaling laws are significantly violated and that the anomalous dimension is negative in many cases provides strong indications that the transitions in frustrated magnets are most probably of very weak first order. We then review the various perturbative and early nonperturbative approaches used to investigate these systems. We argue that none of them provides a completely satisfactory description of the three-dimensional critical behavior. We then recall the principles of the nonperturbative approach -the effective average action method -that we have used to investigate the physics of frustrated magnets. First, we recall the treatment of the unfrustrated -O(N ) -case with this method. This allows to introduce its technical aspects. Then, we show how this method unables to clarify most of the problems encountered in the previous theoretical descriptions of frustrated magnets. Firstly, we get an explanation of the long-standing mismatch between different perturbative approaches which consists in a nonperturbative mechanism of annihilation of fixed points between two and three dimensions. Secondly, we get a coherent picture of the physics of frustrated magnets in qualitative and (semi-) quantitative agreement with the numerical and experimental results. The central feature that emerges from our approach is the existence of scaling behaviors without fixed or pseudo-fixed point and that relies on a slowing-down of the renormalization group flow in a whole region in the coupling constants space. This phenomenon allows to explain the occurence of generic weak first order behaviors and to understand the absence of universality in the critical behavior of frustrated magnets.

We study the optimization of nonperturbative renormalization group equations truncated both in fields and derivatives. On the example of the Ising model in three dimensions, we show that the Principle of Minimal Sensitivity can be unambiguously implemented at order ∂ 2 of the derivative expansion. This approach allows us to select optimized cutoff functions and to improve the accuracy of the critical exponents ν and η. The convergence of the field expansion is also analyzed. We show in particular that its optimization does not coincide with optimization of the accuracy of the critical exponents.

On the example of the three-dimensional Ising model, we show that nonperturbative renormalization group equations allow one to obtain very accurate critical exponents. Implementing the order ∂ 4 of the derivative expansion leads to ν = 0.632 and to an anomalous dimension η = 0.033 which is significantly improved compared with lower orders calculations.Many problems in high-energy as well as in statistical physics call for nonperturbative methods. On the one hand, several physical systems are described by field theories in their strong coupling regime so that the usual perturbative techniques become troublesome. They fail either because only the first orders of perturbation are computed and do not suffice, or because, even when high orders are known, standard resummation techniques do not provide converged results. On the other hand, some phenomena such as confinement in QCD or phase transitions induced by topological defects are genuinely nonperturbative.Apart from some methods restricted to specific dimensions or situations, very few nonperturbative techniques are available. During the last years, the Wilson approach 1 to the renormalization group (RG) has been turned into an efficient tool 2,3,4 . This nonperturbative RG can be implemented in very general situations and, in particular, in any dimension, so that it has allowed one to study several issues difficult to tackle within a perturbative framework among which the three-dimensional Gross-Neveu model 5 , frustrated magnets 6 , the randomly dilute Ising model 7 , and the Abelian Higgs model 8 .This method relies on a nonperturbative renormalization of the effective action Γ, i.e. the Gibbs free energy. It consists in building an effective action Γ k at the running scale k by integrating out only fluctuations greater than k. At the scale k = Λ, Λ −1 denoting the spacing of the underlying lattice, Γ k coincides with the Hamiltonian H since no fluctuation has yet been taken into account while, at k = 0, it coincides with the standard effective action Γ since all fluctuations have been integrated out. Thus, Γ k continuously interpolates between the microscopic Hamiltonian H and the free energy Γ. The running effective action Γ k follows an exact equation which controls its evolution with the running scale k 2 :where t = ln(k/Λ) and Γ(2) k [φ] is the second functional derivative of Γ k with respect to the field φ(q). In Eq. (1), R k (q) is an infrared cutoff function which suppresses the propagation of the low-energy modes without affecting the high-energy ones.Although exact, Eq. (1) is a functional partial integrodifferential equation which cannot be solved exactly. To handle it, one has to truncate Γ k . A natural and widely used truncation is the derivative expansion, which consists in expanding Γ k in powers of ∂φ, keeping only the lowest order terms. Physically, this truncation rests on the assumption that the long-distance physics of a given model is well described by the lowest derivative terms, the higher ones corresponding to less relevant operators. U...

No abstract

Frustrated magnets are a notorious example where the usual perturbative methods are in conflict. Using a nonperturbative Wilson-like approach, we get a coherent picture of the physics of Heisenberg frustrated magnets everywhere between d = 2 and d = 4. We recover all known perturbative results in a single framework and find the transition to be weakly first order in d = 3. We compute effective exponents in good agreement with numerical and experimental data.PACS No: 75.10.Hk, 11.10.Hi, 11.15.Tk Understanding the effect of competing interactions in three dimensional classical spin systems is one of the great challenges of condensed matter physics. However, after twenty five years of investigations, the nature of the universality class for the phase transition of the simplest frustrated model, the antiferromagnetic Heisenberg model on a triangular lattice (AFHT model), is still a strongly debated question [1] . Due to frustration, the ground state of the AFHT model is given by a canted configuration -the famous 120 • structure -that implies a matrix-like order parameter [2] and thus, the possibility of a new universality class. Experiments performed on materials supposed to belong to the AFHT universality class display indeed exponents different from those of the standard O(N ) universality class: for VCl These results however call for several comments. First, the exponents violate the scaling relations, at least by two standard deviations. Second, they differ significantly from those obtained by Monte Carlo (MC) simulations performed either directly on the AFHT model (ν ≃ 0.59(1), γ ≃ 1.17(2), β ≃ 0.29(1), α ≃ 0.24(2)), and on models supposed to belong to the same universality class: AFHT with rigid constraints (ν = 0.504(10), γ = 1.074(29), β = 0.221(9), α = 0.488(30)), dihedral (i.e. V 3,2 Stiefel) models (ν ≃ 0.51(1), γ ≃ 1.13(2), β ≃ 0.193(4), α ≃ 0.47(3)). See Ref.[9] for a review, and references therein. Finally, the anomalous dimensions η obtained by means of scaling relations is found to be negative in experiments as well as in MC simulations, a result forbidden by first principles for second order phase transitions [10] . All these results are hardly compatible with the assumption of universality. It has been proposed that the exponents are, in fact, effective exponents characterizing a very weakly first order transition, the so-called "almost second order phase transition [11][12][13] ".From the theoretical point of view the situation is also very unsatisfactory since one does not have a coherent picture of the expected critical behaviour of the AFHT model between two and four dimensions. On the one hand, the weak coupling expansion performed on the suitable Landau-Ginzburg-Wilson (LGW) model in the vicinity of d = 4 leads to a first order phase transition due to the lack of a stable fixed point [14][15][16] . On the other hand, the low temperature expansion performed around two dimensions on the Non-Linear Sigma (NLσ) model predicts a second order phase transition of the standard O(4)/O(3) universal...

We present a simple approximation of the nonperturbative renormalization group designed for the Kardar-Parisi-Zhang equation and show that it yields the correct phase diagram, including the strong-coupling phase with reasonable scaling exponent values in physical dimensions. We find indications of a possible qualitative change of behavior around d=4. We discuss how our approach can be systematically improved.

No abstract